The Construction of P-Tables and Applications

Authors:Luo QuanComments: 11 Pages. the author is an independent researcher in number theory. This article might include mistakes.

In this article we construct a sequence of p-tables for each prime number p and list out the p-tables for p = 3,5,7. Then we use these p-tables to study congruence equations. Some results about a spacial
family of congruence equations, called combinatorial congruence equations, have been obtained. Finally, the Twin Primes Conjecture and the Goldbach Conjecture are discussed in language of p-tables.
Category:Number Theory

Two Conjectures that Relates Any Poulet Number by a Type of Triplets Respectively of Duplets of Primes

In one of my previous papers I conjectured that there exist an infinity of Poulet numbers which can be written as the sum of three primes of the same form from the following eight ones: 30k+1, 30k+7, 30k+11, 30k+13, 30k+17, 30k+19, 30k+23, 30k+29. In this paper I conjecture that any Poulet number not divisible by 5 can be written as a sum of three primes of the same form from the following four ones: 30k+1, 30k+3, 30k+7 or 30k+9 respectively as a sum from a prime and the double of the another one, both primes having the same form from the four ones mentioned above. Finally, I yet made any other two related conjectures about two types of squares of primes.
Category:Number Theory

A Type of Primes that Seem to Lead to Sequences of Infinite Poulet Numbers in a Recurrent Formula

Though I discovered a lot of sequences of Poulet numbers based on different types of formulas, I never succeded to find a recurrent formula able to produce a subset of Poulet numbers...until now, when I incidentally noticed an interesting relation between two Poulet numbers divisible by 73. Extrapolating the result I obtained a recurrent formula based on primes of the form 30k+13 that seem to lead often to possible infinite sequences of Poulet numbers.
Category:Number Theory

A Very Interesting Formula of a Subset of Poulet Numbers Involving Consecutive Powers of a Power of Two

I studied Poulet numbers for some time but I’m still amazed by the wealth of the patterns that this set of numbers offers; it’s like everything that the prime numbers, in their stubbornly to let themselves understood and disciplined, refuse us, these exceptions of the Fermat’s “little theorem” allow us. This paper states a conjecture about a new subset of Poulet numbers that I discovered by chance.
Category:Number Theory

Four Conjectures About Three Subsets of Pairs of Twin Primes

In this paper are stated four conjectures about three subsets of pairs of twin primes, i.e. the pairs of the form (p^2 + q – 1, p^2 + q + 1), where p and q are primes (not necessarily distinct), the pairs of the form (p + q – 1, p + q + 1), where p, q and q + 2 are all three primes and the pairs of the form (p^2 + q – 1, p^2 + q + 1), where p, q and q + 2 are all three primes.
Category:Number Theory

I evaluate the constant 1/π using the Babylonian identity and complete elliptic integral of first kind. This resulted in two representations in terms of the Euler’s gamma functions and summations.
Category:Number Theory

Proof of the Fermat's Last Theorem

The theorem is proved by means of general algebra. It is based on deduced polynomials a=uwv+v^n; b=uwv+w^n; c=uwv+v^n+w^n and their modifications required to satisfy equation a^n+b^n=c^n. The equation also requires existence of positive integers u_p and c_p such that a+b is divisible by (u_p)^n and c is product of (u_p)(c_p). Based on these conclusions two versions of proof are developed. One of them reveals that after long division of two divisible by c polynomials obtained remainder is coprime with it. In another version transformation of a^n+b^n into expression that allows to apply the Eisenstein’s criterion reveals a contradiction.
Category:Number Theory

Ten Conjectures About Certain Types of Pairs of Primes Arising in the Study of 2-Poulet Numbers

There are many interesting, yet not studied enough, properties of Poulet numbers. In particular, the study of the 2-Poulet numbers appears to be most seductive because in their structure are found together three of the most important concepts in number theory: those of primes, semiprimes and pseudoprimes. In this paper we make few conjectures about primes or pairs of primes, including twin primes, that could be associated to the pairs of primes represented by the two prime factors of a 2-Poulet number.
Category:Number Theory

Seventeen Sequences of Poulet Numbers Characterized by a Certain Set of Smarandache-Coman Divisors

In a previous article I defined the Smarandache-Coman divisors of order k of a composite integer n with m prime factors and I sketched some possible applications of this concept in the study of Fermat pseudoprimes. In this paper I make few conjectures about few possible infinite sequences of Poulet numbers, characterized by a certain set of Smarandache-Coman divisors.
Category:Number Theory

The Smarandache-Coman Divisors of Order K of a Composite Integer N with M Prime Factors

We will define in this paper the Smarandache-Coman divisors of order k of a composite integer n with m prime factors, a notion that seems to have promising applications, at a first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers.
Category:Number Theory

The Cyclic Variation in the Density of Primes in the Intervals Defined by the Fibonacci Sequence

The Riemann R-function can be used to estimate the number of primes in an interval, where its accuracy is affected by the interval to which it is applied. Here, the successive intervals defined by the Fibonacci sequence will be shown to cause more cycles of R-function over- and under-estimation of primes than any of a large landscape of related sequences (calculations were continued up to one billion). The size of this landscape suggests that a special relationship exists between the Fibonacci sequence and the distribution of primes.
Category:Number Theory